When \(\sigma = 1\)  and \(\mathbf {w}\)  is a Brownian motion (or fractional Brownian motion), one can see (in [1]} and [2]} for example) that the previous averaged field enjoys better regularity properties (in space) than \(b\) . On can also consult [3]} and [4]} for a systematic study of the previous averaged field in a more general context.
